Solution of the Sir Epidemic Model for the Spread of Tuberculosis Using the Fourth Order Runge-Kutta and Milne Method

Tuberculosis (TB) is an infectious disease of the human respiratory tract caused by the bacterium Mycobacterium tuberculosis (Mtb). The bacteria that cause TB are a type of bacillus bacteria that are very strong, so it takes a long time to treat this TB disease. This research is a literature study examining the mathematical model of SIR in TB disease. This research involves several stages, including the numerical integration of the SIR model, converting the resulting model into a computer programming language, performing numerical simulations, and observing solution graphs. This study aims to solve the SIR model of tuberculosis transmission using the fourth-order Runge-Kutta method and the Milne method. The resulting SIR model is a nonlinear differential equation model. The object of research in this study is the SIR Mathematical Model. The procedure for creating the SIR mathematical model consists of seven steps: case identification, establishing assumptions, creating the mathematical model, model analysis, model interpretation, model validation, and using the model. The research method employed is a literature study approach with a numerical component. Simulations were carried out twice for each method. The results of the numerical simulation in the MATLAB program show that both methods produce solutions with similar behaviour. However, in theory, the Milne method has a higher level of accuracy than the fourth-order Runge-Kutta method. The graph also shows that a population/individual suffering from tuberculosis will recover over time, assuming they undergo treatment or adopt a healthy lifestyle. The infection population will experience a decline towards an equilibrium point as time passes.